12.2 Percentages
Key Takeaways
- A percent of a number is the decimal times the number: 15% of 80 is 0.15 × 80 = 12, and 18 is 40% of 45 because 18/45 = 0.4.
- Percent increase multiplies by 1 + p; decrease multiplies by 1 − p. $40 up 15% is $46; $80 off 25% is $60.
- To recover the original amount, divide by the factor you multiplied by. After a 20% discount, $48 means $48 / 0.8 = $60, not $48 × 1.20.
- Stacked percents multiply: 20% off then another 20% off a $100 jacket is $64 (36% off), not a single 40% off to $60.
- Absolute change and percent change are different: +20 students is 5% of 400 but 20% of 100, and +5% then −5% does not return 2,400 people to 2,400 (it lands on 2,394).
Percentages are the second PSD skill/knowledge testing point on PSAT 8/9. Independent OpenExamPrep teaching for this skill uses original numbers and school-and-store contexts; it is not an official College Board worksheet. A calculator is allowed on every Math item. SPR answers that are percents are usually the number without a percent sign: if the value is 40%, type 40. Do not type 40% or 0.40 unless the stem asks for a decimal.
Percent of a number
n% of a number means (n/100) times that number. Move the decimal two places left, then multiply.
| Percent | Decimal | Example |
|---|---|---|
| 1% | 0.01 | 1% of 80 = 0.80 |
| 5% | 0.05 | 5% of 80 = 4 |
| 10% | 0.10 | 10% of 250 = 25 |
| 12.5% | 0.125 | 12.5% of 80 = 10 |
| 15% | 0.15 | 15% of 80 = 12 |
| 20% | 0.20 | 20% of 45 = 9 |
| 25% | 0.25 | 25% of 64 = 16 |
| 50% | 0.50 | 50% of 90 = 45 |
| 75% | 0.75 | 75% of 80 = 60 |
| 100% | 1 | 100% of 40 = 40 |
| 150% | 1.50 | 150% of 40 = 60 |
Worked: 15% of 80.
0.15 × 80 = 12. Check: 10% of 80 is 8, and 5% is 4, so 15% is 8 + 4 = 12.
Worked: 25% of 64.
0.25 × 64 = 16. Check: 25% is one-fourth, and 64/4 = 16.
What percent is one number of another?
The proportion is (part)/(whole) = (percent)/100, or simply divide part by whole and convert to a percent.
18 is what percent of 45?
18/45 = 0.4 = 40%.
Cross-multiply: 18/45 = p/100, so 18 × 100 = 45p, 1800 = 45p, p = 40.
12 is what percent of 80?
12/80 = 0.15 = 15%.
9 is what percent of 30? 9/30 = 0.3 = 30%.
The whole is the number after of. Switching part and whole is the usual error: 45/18 ≈ 2.5 would be 250%, which answers a different question (45 is 250% of 18).
Percent increase and percent decrease
A percent increase of p means multiply the original by 1 + p (as a decimal). A percent decrease of p means multiply by 1 − p.
A jacket costs $40, then the price rises 15%.
Increase amount: 0.15 × 40 = $6. New price: 40 + 6 = $46, or 40 × 1.15 = 46.
A backpack costs $80 and is 25% off.
Discount: 0.25 × 80 = $20. Sale price: 80 − 20 = $60, or 80 × 0.75 = 60.
A score goes from 40 to 50. Absolute change is +10 points. Relative change is 10/40 = 0.25 = 25% increase. A score from 80 to 90 is also +10 points, but 10/80 = 0.125 = 12.5% increase. Same absolute change, different percent change.
A price falls from $80 to $68. Drop of $12; 12/80 = 0.15 = 15% decrease. Check: 80 × 0.85 = 68.
A price rises from $50 to $65. Rise of $15; 15/50 = 0.30 = 30% increase. Check: 50 × 1.30 = 65.
Original amount before a percent change
When the stem gives the new amount and the percent, divide by the factor you would have multiplied by. Do not add or subtract the percent of the new amount; that percent belonged to the unknown original.
After a 25% increase, a price is $60. What was the original?
Original × 1.25 = 60
Original = 60 / 1.25 = 48
Check: 25% of 48 is 12, and 48 + 12 = 60.
After a 20% discount, a shirt costs $48. Original?
Original × 0.80 = 48
Original = 48 / 0.80 = 60
Check: 20% of 60 is 12, and 60 − 12 = 48.
Trap: 48 × 1.20 = 57.60. That adds 20% to the sale price. The 20% was taken from a larger number, so you have to divide, not multiply.
After a 10% decrease, enrollment is 450. Original x satisfies x × 0.90 = 450, so x = 450 / 0.90 = 500. Check: 10% of 500 is 50, and 500 − 50 = 450.
Stacked percents do not add
20% off, then another 20% off, is not 40% off. Percents of different bases multiply.
A $100 jacket, 20% off: 100 × 0.80 = $80. Then another 20% off the sale price: 80 × 0.80 = $64.
Total dollars off: $36, which is 36% of 100, not 40%. A single 40% off would be 100 × 0.60 = $60. The extra 20% was 20% of 80, which is $16, not another $20.
Same idea with 20% off then 10% off: 100 × 0.80 × 0.90 = $72. That is 28% off, not 30%. A single 30% off would be $70.
Markup then markdown: a store marks a $40 item up 50% to 40 × 1.50 = $60, then takes 50% off 60, which is $30. You do not return to $40, because 50% of 60 is $30, while 50% of 40 was $20.
Relative versus absolute
School A has 400 students and adds 20. Relative change: 20/400 = 5%. School B has 100 students and adds the same 20. Relative change: 20/100 = 20%.
The absolute change is 20 in both buildings. The percent change is not. A stem that asks which school grew faster as a percent wants School B. A stem that asks which added more students wants a tie.
+5% then −5% is not a wash. A town of 2,400 grows 5%: 2400 × 1.05 = 2,520 (increase of 120). Then it falls 5% from 2,520: 5% of 2,520 is 126, and 2520 − 126 = 2,394. Factor check: 1.05 × 0.95 = 0.9975, and 2400 × 0.9975 = 2394. The down 5% used a larger base, so it removed more people than the up 5% added.
Percent of a percent
Take the percent of the subgroup, not the percent of the original whole, unless the stem says otherwise.
A grade has 80 students. 40% take art: 0.40 × 80 = 32 art students. Of those art students, 25% take pottery: 0.25 × 32 = 8 pottery students. As a percent of the whole grade: 8/80 = 10%, which is also 0.40 × 0.25 = 0.10.
Wrong moves: 40% − 25% = 15%, or 40% + 25% = 65%. Those mix two different wholes.
Another: 30% of 80 students play soccer: 0.30 × 80 = 24. Then 25% of those soccer players are eighth graders: 0.25 × 24 = 6.
Tax and tip: read which amount the percent uses
A meal is $80. Sales tax is 10%: 0.10 × 80 = $8, so food plus tax is $88. A 15% tip on the food only is 0.15 × 80 = $12, so the grand total is 88 + 12 = $100.
If the stem says the 15% tip is on the after-tax amount, the tip is 0.15 × 88 = $13.20, and the grand total is 88 + 13.20 = $101.20. The arithmetic is not hard; the trap is applying 15% to the wrong base. Copy the sentence onto scratch paper and underline the words after tax or before tax.
Calculator and SPR habits
Desmos will compute 48 / 0.8 = 60 instantly after you have written Original × 0.8 = 48. It will also compute 48 × 1.2 = 57.6 if you feed it the trap. The setup is the scored skill. On multiple choice, plugging an original-price candidate forward (take 20% off and see whether you land on $48) is a legal check. On SPR, that check is how you catch a 57.6 before you type it. Type 60, not $60.
A backpack costs $80. During a sale, the price is reduced by 25%. What is the sale price in dollars?
After a 20% discount, a shirt costs $48. What was the original price in dollars?
A $100 jacket is marked 20% off. Then the sale price is reduced by another 20%. What is the final price in dollars?